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Great write-up. Inspired me to try how much further ICL could go beyond "simpler" mappings (OP shows pretty nice results for two linear and two quadratic functions). As such, I tried a damped sinusoid: 

with the prompt:

x=3.984, y=6.68
x=2.197, y-2.497
x=0.26, y=-7.561
x=6.025, y=-1.98
x=7.126, y=-4.879
x=8.584, y=-0.894
x=9.97, y=3.403
x=11.1, y=2.45
x=12.09, y=-0.452
x=13.72, y=-2.48
x=14.81, y=-0.606
x=10, y=

but didn't get any luck. Maybe I need more points, especially around the troughs and valleys.